6.3 Modeling Linear Associations

Example: Using a Line to Model an Association The mean ticket prices for the top-50-grossing concert tours are shown in for various years. Let p be the mean ticket price (in dollars) at t years since 1995.

Example: Continued 1. Construct a scatterplot of the data. 2. Is there a linear association, a nonlinear association, or no association? 3. Compute r. On the basis of r and the scatterplot, determine the strength of the association. 4. Draw a line that comes close to the points of the scatterplot.

Solution 1. First, we list values of t and p in the table below. For example, t = 3 represents 1998, because 1998 is 3 years after 1995; and t = 6 represents 2001, because 2001 is 6 years after 1995.

Solution Next, we sketch a scatterplot. Because t is the explanatory variable and p is the response variable, we let the horizontal axis be the t-axis and let the vertical axis be the p-axis.

Solution 2. The points appear to lie very close to a line, so the association is linear. 3. We use a TI–84 to compute r  ≈ 0.99 . Because r is so close to 1 and because the points in the scatterplot appear to lie so close to a line, we conclude that the association is very strong.

Solution We sketch a line that comes close to the points of the scatterplot. The line needn’t contain any of the points, but it should come close to all of them.

Solution We sketched a line that describes the mean-ticket-price situation. However, this description is not exact. For example, the line does not describe exactly what happened in the years 1998, 2001, 2004, 2008, or 2011, because the line does not contain any of the data points. However, the line does come pretty close to these data points, so it suggests pretty good approximations for those years. The process of choosing a line to represent the association between time and mean ticket prices is an example of modeling.

Definitions Model A model is a mathematical description of an authentic situation. We say the description models the situation. Linear Model A linear model is a nonvertical line that describes the association between two quantities in an authentic situation

Example: Using a Linear Model to Make Estimates 1. Use the linear model shown to estimate the mean ticket price in 2000. 2. Use the linear model to estimate the mean ticket price in 2010. 3. Use the model to estimate in which year the mean ticket price was $65.

Solution 1. The year 2000 corresponds to t = 5, because 2000 – 1995 = 5. To find the mean ticket price in 2000, we locate the point on the linear model where the t-coordinate is 5 (see the blue arrows). The p-coordinate of that point is 42. So, according to the model, the mean ticket price in 2000 was approximately $42.

Solution 1. We verify our work by checking that our result is consistent with the values shown in the table below. Since the mean ticket price was $33 in 1998 and $47 in 2001, it follows that the mean ticket price in 2000 probably would be between $33 and $47, which checks with our result of $42.

Solution 2. The year 2010 corresponds to t = 15, because 2010 – 1995 = 15. To find the mean ticket price in 2010, we locate the point on the linear model where the t-coordinate is 15 (see the red arrows). The p-coordinate of that point is 79. So, according to the model, the approximate mean ticket price in 2010 was $79. This result is consistent with the values in the table.

Solution 3. To find the year when the mean ticket price was $65, we locate the point on the linear model where the p-coordinate is 65 (see the green arrows). The approximate t-coordinate of that point is 11.2. So, according to the linear model, the mean ticket price in about 1995 + 11 = 2006 was $65. This result is consistent with the values in the table.

Input, Output Definition An input is a permitted value of the explanatory variable that leads to at least one output, which is a permitted value of the response variable.

Interpolation, Extrapolation Definition For a situation that can be described by a model whose explanatory variable is x, • We perform interpolation when we use a part of the model whose x-coordinates are between the x-coordinates of two data points. • We perform extrapolation when we use a part of the model whose x-coordinates are not between the x-coordinates of any two data points.

Model Breakdown Definition When a model gives a prediction that does not make sense or an estimate that is not a good approximation, we say model breakdown has occurred.

Intercepts of a Line Definition An intercept of a line is any point where the line and an axis (or axes) of a coordinate system intersect. There are two types of intercepts of a line sketched: • An x-intercept of a line is a point where the line and the x-axis intersect. The y-coordinate of an x-intercept is 0. • A y-intercept of a line is a point where the line and the y-axis intersect The x-coordinate of a y-intercept is 0.

Example: Finding Intercepts and Coordinates 1. Find the x-intercept of the line. 2. Find the y-intercept of the line. 3. Find y when x = 4. 4. Find x when y = –2.

Solution 1. The line and the x-axis intersect at (–2, 0). So, the x-intercept is (–2, 0). 2. The line and the y-axis intersect at (0, 1). So, the y-intercept is (0, 1). 3. The blue arrows show that the input x = 4 leads to the output y = 3. So, y = 3 when x = 4. 4. The red arrows show that the output y = –2 originates from the input x = –6. So, x = –6 when y = –2.

Example: Intercepts of a Model The percentages of cell phone users who send or receive text messages multiple times per day are shown in for various age groups. Let p be the percentage of cell phone users at age a years who send or receive text messages multiple times per day.

Example: Continued 1. Draw a model that describes the association between a and p. 2. Predict the percentage of 35-year-old cell phone users who send or receive text messages multiple times per day. Did you perform interpolation or extrapolation? 3. Find the p-intercept. What does it mean in this situation? Did you perform interpolation or extrapolation? 4. Find the a-intercept. What does it mean in this situation? Did you perform interpolation or extrapolation?

Solution 1. We begin by viewing the positions of the data points in the scatterplot It appears a and p are linearly associated, so we sketch a line that comes close to the data points. (see Fig. 90).

Solution 2. We locate the point on the linear model where the a- coordinate is 35 (see the green arrows). The p- coordinate of that point is 54. So, 54% of 35-year- old cell phone users send or receive text messages multiple times per day. We have performed interpolation because the a-coordinate 35 is between the a-coordinates of the data points (29.5, 63) and (39.5, 42).

Solution 3. The p-intercept is (0, 106), or p = 106, when a = 0. According to the model, 106% of newborns who use cell phones send or receive text messages multiple times per day. We have performed extrapolation because the a-coordinate 0 is not between the a-coordinates of two data points. Model breakdown has occurred for two reasons: Percentages cannot be larger than 100% in this situation, and newborns cannot send or receive text messages.

Solution 4. The a-intercept is (71, 0), or p = 0, when a = 71. According to the model, no 71-year-old cell phone users send or receive text messages multiple times per day. We have performed extrapolation because the a-coordinate 71 is not between the a-coordinates of two data points. Model breakdown has occurred because some 71-year-old cell phone users send or receive text messages multiple times per day.

Example: Modifying a Model Additional research yields the data shown in the first and last rows of the table. Use this data and the following assumptions to modify the model we found in the previous example.

Example: Continued Children 3 years old and younger do not send or receive text messages multiple times per day. • The percentage of cell phone users who send or receive text messages levels off at 5% for users over 80 years in age. • The age of the oldest cell phone users is 116 years.

Solution Recall that p is the percentage of cell phone users at age a years who send or receive multiple text messages per day. We sketch a scatterplot of the data in table, and taking into account the three assumptions, we draw a model that comes close to the data points